Cotangent space

In differential geometry, the cotangent space is a vector space associated with a point ${\displaystyle x}$ on a smooth (or differentiable) manifold ${\displaystyle {\mathcal {M}}}$; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, ${\displaystyle T_{x}^{*}\!{\mathcal {M}}}$ is defined as the dual space of the tangent space at ${\displaystyle x}$, ${\displaystyle T_{x}{\mathcal {M}}}$, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.